Abelian Fractional Quantum Hall States Research Articles

Quantum Hall valley ferromagnets as a platform for topologically protected quantum memory

Materials hosting topologically protected non-Abelian zero modes offer the exciting possibility of storing and manipulating quantum information in a manner that is protected from decoherence at the hardware level. In this paper, we study the possibility of realizing such excitations along line defects in certain fractional quantum Hall states in multivalley systems. Such line defects have been recently observed experimentally between valley polarized Hall states on the surface of Bi(111), and excitations near these defects appear to be gapped (gapless) depending on the presence (absence) of interaction-induced gapping perturbations constrained by momentum selection rules, while the position of defects is determined by strain. In this paper, we use these selection rules to show that a hybrid structure involving a superlattice imposed on such a multivalley quantum Hall surface realizes non-Abelian anyons, which can then be braided by modulating strain locally to move line defects. Specifically, we explore such defects in Abelian fractional quantum Hall states of the form $\ensuremath{\nu}=2/m$ using a K-matrix approach, and identify relevant gapping perturbations. Charged modes on these line defects remain gapped, while charge neutral valley pseudospin modes may be gapped with the aid of two (mutually orthogonal) superlattices, which pin noncommuting fields. When these superlattices are alternated along the line defect, non-Abelian zero modes result at points where the gapping perturbation changes. Given that these pseudospin modes carry no net physical charge or spin, the setup eschews utilizing superconducting and magnetic elements to engineer gapping perturbations. We provide a scheme to braid these modes using strain modulation, and confirm that the resulting unitaries satisfy a representation of the braid group.

Parafermions in hierarchical fractional quantum Hall states

This work establishes the nature of non-Abelian parafermion defects, which are localized on domain walls at the interface of hierarchical Abelian fractional quantum Hall states. The author uncovers a sequence of parafermions and show that their quantum dimensions are directly related to the topological properties of the bulk Abelian phase, thus revealing hierarchical fractional quantum Hall states as a setting for the realization of non-Abelian defects.

Paired parton quantum Hall states: A coupled wire construction

The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a new family of incompressible paired parton FQH states at filling $\nu=1/6$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\nu=1/3$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-like anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principle existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a new notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimension.

Solvable models for neutral modes in fractional quantum Hall edges

We describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode---i.e., an edge mode that does not carry electric charge. These models consist of two counterpropagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the $\ensuremath{\nu}=2/3$ state is the most prominent example.

Ground-state degeneracy of topological phases on open surfaces.

We relate the ground state degeneracy of a non-Abelian topological phase on a surface with boundaries to the anyon condensates that break the topological phase into a trivial phase. Specifically, we propose that gapped boundary conditions of the surface are in one-to-one correspondence with the sets of condensates, each being able to completely break the phase, and we substantiate this by examples. The ground state degeneracy resulting from a particular boundary condition coincides with the number of confined topological sectors due to the corresponding condensation. These lead to a generalization of the Laughlin-Tao-Wu charge-pumping argument for Abelian fractional quantum Hall states to encompass non-Abelian topological phases, in the sense that an anyon loop of a confined anyon winding a nontrivial cycle can pump a condensed anyon from one boundary to another. Such generalized pumping may find applications in quantum control of anyons, eventually realizing topological quantum computation.

Bulk-edge correspondence in (2 + 1)-dimensional Abelian topological phases

The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at $\nu=8$ and 12, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\nu=8/7, 12/11, 8/15, 16/5$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a $p^n$-th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $U(1)^N$ Chern-Simons theory parameterized by a K-matrix.

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in case of integer quantum Hall effect, for fractional quantum Hall states. In case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.

Screening properties and phase transitions in unconventional plasmas for Ising-type quantum Hall states

Utilizing large-scale Monte-Carlo simulations, we investigate an unconventional two-component classical plasma in two dimensions which controls the behavior of the norms and overlaps of the quantum-mechanical wavefunctions of Ising-type quantum Hall states. The plasma differs fundamentally from that which is associated with the two-dimensional XY model and Abelian fractional quantum Hall states. We find that this unconventional plasma undergoes a Berezinskii-Kosterlitz-Thouless phase transition from an insulator to a metal. The parameter values corresponding to Ising-type quantum Hall states lie on the metallic side of this transition. This result verifies the required properties of the unconventional plasma used to demonstrate that Ising-type quantum Hall states possess quasiparticles with non-Abelian braiding statistics.

Clustering Properties and Model Wave Functions for Non-Abelian Fractional Quantum Hall Quasielectrons

We present model wave functions for quasielectron (as opposed to quasihole) excitations of the unitary Z_{k} parafermion sequence (Laughlin, Moore-Read, or Read-Rezayi) of fractional quantum Hall states. We uniquely define these states through two generalized clustering conditions: they vanish when either a cluster of k+2 electrons is put together or when two clusters of k+1 electrons are formed at different positions. For Abelian fractional quantum Hall states (k=1), our construction reproduces the Jain quasielectron wave function and elucidates the difference between the Jain and Laughlin quasielectrons.

Model of Statistically Coupled Chiral Fields on the Circle

Starting from a field theoretical description of multicomponent anyons with mutual statistical interactions in the lowest Landau level, we construct a model of interacting chiral fields on the circle, with the energy spectrum characterized by a symmetric matrix gαβ with nonnegative entries. Being represented in a free form, the model provides a field theoretical realization of (ideal) fractional exclusion statistics for particles with linear dispersion, with gαβ as a statistics matrix. We derive the bosonized form of the model and discuss the relation to the effective low-energy description of the edge excitations for Abelian fractional quantum Hall states in multilayer systems.

Classification of Abelian quantum Hall states and matrix formulation of topological fluids.

We give a simple and unified treatment of quantum topological fluids such as the quantum Hall fluid. We show that the order in such fluids can be characterized by a symmetric matrix K, in terms of which various physical quantities can be determined. We construct K by a matrix iteration procedure which may be decomposed into two simple elementary steps. The hierarchy construction is shown to be contained in our matrix iteration construction. The relationship between the vortex basis and the dual electron basis is clarified. We also show that under certain mild assumptions the generalized hierarchy construction exhausts all possible Abelian fractional quantum Hall states. We identify and determine the topological quantity known as the shift. Our formalism may be relevant for recent experimental data on multilayered systems.